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OPTIMAL ORDERING REGIONS UNDER JOINT REPLENISHMENT by Donald B. Rosenfield Visiting Associate Professor M.I.T. Sloan School of Management Senior Consultant Arthur D. Little, Inc. Working Paper #1661-85 May 1985 IP_I 1_ 1--s---1)-11__ 1. Introduction The inventory theory literature contains extensive treatment of as treatment of joint replenishment single item policies, as well (i.e. treated very thoroughly is the problem of joint is, a single set-up cost for two or more items, demand. optimal (r, What is not ordering frequency) under deterministric demand. 2] Ignall replenishment, that under stochastic policies may be complex. Johnson 3] and Kalin s) policies under certain conditions where ordering region. and that notes that this problem is difficult, Silver 6] [4] is an optimal hypothesizes can order regions conditions of major and minor set-up costs. identify under the There are no definitive results, however, about the nature of the optimal ordering region This paper deals with the most basic type of replenishment: and when to order commonly faced with this problem. however, will also for the two items and Indeed, pratitioners are Typically items are ordered when reached for either item. set order points when reaches a given order some theoretical basis, as Some pratitioners, the weighted level. sum of stock This approach has the development in the remainder The development demonstrates the nature of paper shows. order surfaces for a single period. not shown Both the theoretical implications are self-evident. an order point is issue in joint two items under a common set-up charge independent stochastic demands. practical of the the optimal While the analogous results are for multiple periods or the infinite horizon, the single- period results should indicate . the nature of the solution. 1 11 Based on results from single-item theory, under and holding costs, when items are ordered, order up to. Hence with two independent ordered, there issue is the determination of it exists) 2. is a unique vector level the linear penalty there are unique levels items, when (s 1 , s2) items are to order up to. surface on the cartesian space from which it is optimal to The (if to order. Analysis Consider a family of two items with a joint order cost. Thus the cost of ordering is c(x,y) = k + c x + c2 y if either x > 0 or y > 0 = 0 otherwise where x = order for item 1 y = order for item 2 k = setup cost The issue becomes the costs of alternative policies for a given inventory position (xo,yo). the two items expected costs order-up-to of h For linear holding and penalty costs for and h 2 and Pi given an initial level and P2 respectively, inventory level (x,y) is + c2(Y - y0 ) k + C1(x - x 0) x + £ Pl( -X) l( )d + £ hl(x- 0D + )dE X p 2 (E-y), 2(E)d y )l( -< X + f h 2 (Y-E)0 2 -a 2 (E)dE (xo, the total yo) and an 1 and where 2 are the two demand densities. If x=xo and Y=YO, then the same expression with k=O holds. Let i.(y) f = where Fi f = (-y)*i(E)dE y (1-Fi(E))dt y is the cumulative distribution for *i. For a normal distribution, for example, y -u Gi(Y) iG( -- = - where pi = mean of Qa = standard deviation of = Go(Y) I i i (-y)(E)dE y where O is a standard normal density function. Additionally, Y - I (Y-E)0i(E)dE = y - f ui+ (i-Y)i(E)dE Given these expressions, we see that if an order is made to bring inventories up to the levels x and y, the cost is k-ClX + 0 (c + (hl 1 - c2Y - hu + (C + P)G 1 (x) + + hl)X - 1 2 h2U 2 + h 2 )Y (h 2 + 2 )G 2 (y) 3 11~--~-. ~_.~ ~11_1_1__~_11~.1^ XI__11_^~ __ I· . ~ _- - ---- _ .- (1) Denoting the function in braces as F(x,y), convex with a unique minimum F(x*, y*) and noting that F(x,y) is we see that the optimal policy is 1) order up to (x*, y*) if x 0 and F(xO, 2) yO) or der up to (x O, and F(x O, yO) or der up to 3) if x0 F(x O , (x*, y) y*) y* x*, YO < y* + k < x*, YO > Y* YO ) + k Do not order otherwise Conditions 2 and 3 yield the usual policy yO < + k > if x0 yO) > F(x*, and F(x o , 4) F(x*, y*) y*) x*, < in y* separable. and x respectively, The interesting case type of order points in an (s, S) as the function in braces is s condition 1, which defines the optimal regions to order when both items are below their optimal levels. The optimal F(xo, = yO) surface F(x*, y*) in the region x 0 two variables x and y, F along y(x) to be zero. So dy 2y dx + aF = 2x 0 or F -d x = d_! dx F From (1), c. ci y the partials are of the form + hi+ i. (Pi 1 + is (2) isocontours (2) can be found by the equating the directional derivative of aF yO < y* + k With a continuous function of of the form < x*, h )Gi i i 4 Furthermore, since Gi(z) f = (E-z)Ol(e)dE I (1-Fi(M))dE = Z Gi(z) = Z - Fi(z) 1 Hence the differential equation for the optimal ordering contour is d2 = - dx (PL + h 1 )Fl(x) l (P2 +h2.)F 2 (Y) (p - (P 2 - c1) c2 ) which yields G 2 (x)(p2 + h 2) + G(x)(p 1 + h1 ) + (c 2 + h 2 )Y + (c1 + hl)x = C (3) where C is a constant. The interpretation of (3) is as follows: The optimal ordering surface is obtained when the sum of functions of x and y are constant. Each function represents the relative "vulnerability" of the inventory for one of the two products. minimized upon ordering). (The functions are When the total vulnerability reaches a threshold, an order is placed. 3. Discussion This type of relationship typically yields a curve of the form shown in Figure 1. When x is near its optimal order point and y is near its optimal order up to level function R 2 (y) = G 2 (y)(p 2 + h 2) + (the point a in Figure 1) the (c 2 + h 2 )y is close to its minimum (that defines the order-up-to level) and is changing very slowly. the other hand, Rl(x) = Gl(x)(pl + h1 ) + (c1 On + hl)x is changing rapidly, as x is near its order point and not hear its minimum. Thus, in order for (3) to hold, small changes in x must be accompanied by large changes in y. The opposite holds at point b. At point c, both functions R 1 (y) and R 2 (x) are changing at about the 5 same rate (when scaled) and hence the optimum curve is pitched at 45 in order to maintain In practice, difficult, of (3). since the use of (3) might be computationally it might make sense to approximate the curve by the union two rays and one line segment arising from the three curves x = constant = x* - Q (4a) y = constant = y* Q2 (4b) - and Q1 x where Q + Q2 = constant (4c) and Q2 are the lot sizes (order-up-to-level minus item order point) for x and y respectively. order points if are the The joint replenishment did not exist. ordering region is the union the curves x* and y* are the The constants in 4a and 4b hence optimal order-up to levels. individual single of three regions below or to the left of 4a through 4c. In other words, there should be separate trigger points for x, y and a weighted sum of x and y. This allows more accuracy than a system with order points for x and y only. While the curve of (3) has not been demonstrated to hold for the infinite horizon or average cost cases, there is a theoretical aggregate sums of basis for trigger levels for stock. As a pratical matter, the weighting on the weighted sum curve should be based on the maximum replensihment quantities for each item (i.e. the lot hundred unit sizes Q1 and Q2)- If, replenishment amount for for example, there is a one- y and a ten-unit amount for x, then y should be more heavily weighted in determining the weighted sum reorder level. This proportional proximity to type of approach examines the sum of the individual 6 item order points. The constant for the weighted sum constraint relationship (3) and is hence the left either * - (x*, Q2) or (x* is the same as C in side of evaluated at - Q, y*). Sometimes, when both items are ordered, smaller (3) set-up for the second item. of what is known as can-order points. This there is an additional leads to the desirability The same type of arguments can be used to show the optimality of the type of policy shown in Figure 2. The curve of the oint order surface between the unconditional order and can-order points is based. on the same type of constant directional derivative argument. 7 11 References 1. R. Brown, Decision Rules for Inventory Management, Rinehart and Winston, NY, 1967. 2. E. Ignall, "Optimal Continuous Review Policies for Two-product Inventory Systems with Joint Setup Costs", Management Science, 15, 1969, 278-283. 3. E. Johnson, "Optimality and Computation of (, S) Policies in the Multi-item Infinite Horizon Inventory Problem", Management Science, 13, No. 7, 1967, 475-491. 4. D. Kalin, "On the Optimality of (, s) olicies", Mathematics of_Operations Research, 5, No. 2, May 1980, 293-307. 5. H. Scarf, "The Optimality of (S, s) Policies in the Dynamic Inventory Problem", Mathematical Methods in the Social Sciences, K. Arrow, S. Karlin, and P. Suppes (eds,), Stanford University Press, Stanford, CA, 1960. 6. E. Silver, "A Control System for Coordinated Inventory Replenishment", International Journal of Production Research, 22, No. 6, 1974, 647-671. 7. E. Silver, "Determining Order Quantities in Joint Replenishment under Deterministic Demand, Management Science, 22, No. 12, 1976, 1351-1361. 8. E. Silver, "Some Characteristics of a Special Joint-order Inventory Model", Operations Research, 13, No. 2, March 1965, 319-327. 9. A. Veinott, Optimal Policy for a Multi-Product Dyanmic Nonstationary Inventory Problem", Management siet sence, 12, Nov. 1965, 206-221. 8 Holt, No. 3, Optimal y1 order up to level for y Optimal y* order point for y alone Optimal order point for x alone x Optimal order up to level for x x2 Figure 1. Optimal Ordering Surface 9 III Optimum order-up-to level - F I y I y can-order point I I joint order surface unconditional y-order point I I I I I x x Can-order point Unconditional x order point Arrows Represent Optimal Orders Figure 2. Optimal ordering with can-order points. 10